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Studying for a test? Prepare with these 9 lessons on Exponential and logarithmic functions.
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VI HART: All right. So I'm Vi Hart and I'm here with Sal Khan and-- SAL KHAN: Hello. VI HART: Yeah. We're talking about just how we think about numbers and what is the most natural way to think about them in our everyday lives. SAL KHAN: And Vi said that she was going to test me right now. VI HART: Yeah. All right, can I borrow the pen? SAL KHAN: Yes. Yes. VI HART: I get to use the official pen and screen-- oh wait. [INAUDIBLE] SAL KHAN: No, that's all screen. Yes. VI HART: OK. OK. SAL KHAN: You need training, Vi. VI HART: I need training. Yay. SAL KHAN: It looks like a pizza. VI HART: It's a triangle. SAL KHAN: Right. Where's your test, Vi? VI HART: OK. SAL KHAN: You're diverging. VI HART: I'm sorry. All right, so here's a number line. Regular old number line. No, wait. I want to start at one. All right, we're going to start at one, and we're going to go all the way to a million. And I'm going to give you the pen now, and I'm going to ask, where is 1,000? SAL KHAN: Where is 1,000? Where is 1,000? I see. I see what you're doing. VI HART: So you can think about this logically. SAL KHAN: Yes. Yes. So I'll tell you what went through my brain. My first knee-jerk reaction was to put 1,000 like right over here. That's what I was tempted to do. VI HART: Mhm. SAL KHAN: And then my brain kicked in. VI HART: Right. Yeah. You think of it-- SAL KHAN: My highly-analytical mind. VI HART: Yeah. Because you know there is a correct answer to this problem. We can think, all right, where is 1,000 on the number line related to a million? Well, a million divided by 1,000. SAL KHAN: Is 1,000. VI HART: You want one thousandth. SAL KHAN: Thousandth. Right. So it's not there. I was going to draw it a tenth of the way. No. 1,000 is like there. You barely notice the difference between that and-- VI HART: Yeah. You couldn't even see the difference in thousandths. SAL KHAN: Yes. So this is fascinating. What is this about? Why did I do that? VI HART: Yeah. Why do we think of 1,000 as being much closer to a million than it is? And we do this, actually, all the time. We're not so used to having to think about the difference between 1,000 and a million. But when we're thinking about the difference between 1 and 2, or the difference between 2 and 3, or 1 and 10, we think, 1 and 2. There's a big difference there. 2 is twice what 1 is. SAL KHAN: Yes. It's twice 1. Right. VI HART: And the difference between 9 and 10 is the same distance when you're looking at it at the usual scale. It's 1. SAL KHAN: Right. Right. VI HART: But when we're thinking about real-life things, well, the difference between 9 and 10 isn't so big in any real life situation. SAL KHAN: No. But the difference between 1 and 2 is huge. VI HART: Yeah. SAL KHAN: In real-life it's double. VI HART: Yes. SAL KHAN: Right. VI HART: So now we have to think on a logarithmic scale is what it is. SAL KHAN: Oh. Yes. The old logarithmic scale. So what you're saying is that we, as humans, even though everything we're taught is these linear scales, where we want to say this is 1, and then maybe this is 10, and then this is 20-- even though that's what we're taught, and that's what most of our mathematics, we plot lines and stuff like that. VI HART: Yeah. That's how we draw it out on paper, usually. SAL KHAN: Yes. VI HART: But that doesn't make sense, usually, for how we think about things. Because the difference between 5 gazillion and 5 gazillion and 10 is-- SAL KHAN: Is nothing. Is nothing. VI HART: Nothing. Whereas the difference between 1 and 10 is huge. SAL KHAN: Right, right, right. And so that's why almost the multiple matters more than the absolute distance between the numbers. VI HART: Yeah. SAL KHAN: Absoluately. And that's what the logarithmic scale captures. VI HART: It is. And that's why we see the logarithmic scale in so many things in real life. As a mathemusician, on the piano we see it. SAL KHAN: Yes. VI HART: It's actually the logarithmic scale. So let's get our piano picture out. SAL KHAN: Oh, look at that. There's a piano. VI HART: Yeah. OK. OK. Can I have the pen? SAL KHAN: Here you go. Yes. VI HART: All right. Let's see if I can figure this out. All right. So here we have this C. Let's call it middle C. And here we have this D. And there's a certain distance between these. And then here is this C and this D. And when we're listening to these notes, we think, all right, they're one note apart. This is the same distance here between here and here and here and there. SAL KHAN: Yes. VI HART: But if you look at the actual frequencies, the distances are not the same. This was maybe a bad example, because I don't know the frequency of D, but-- SAL KHAN: No. Well, we could-- VI HART: Well I'll give you an example I can give numbers to, which is maybe the difference between this octave and the difference between this octave. Right? If this is C-- SAL KHAN: Call it x. VI HART: x. SAL KHAN: Whatever the frequency is. VI HART: Right. Yes. This is great. SAL KHAN: It's like 440 kilohertz. I don't know what it is. VI HART: No. A is 440. SAL KHAN: A is 440. VI HART: C is-- SAL KHAN: Well we'll call it x. VI HART: More like, I don't know, let's say 300. SAL KHAN: Yes. VI HART: All right. So if this is 300 or 300x or just x, then this frequency would be 600. SAL KHAN: 600. Right. It doubles. VI HART: It doubles when you go up an octave. And this would be 1,200 up here, this C. We're in a weird scale here. But the difference between here is 300. And the difference between here is 600. But when we're listening to octaves, we feel like the difference between this octave shouldn't be half as much as the difference between these two notes. Right? The distance from an octave should be an octave. Right? SAL KHAN: Right. So fundamentally the way we perceive pitch is logarithmic. VI HART: It's just fundamentally logarithmic. If you want to have all your notes on the piano be right next to each other, instead of having a piano where you have one key for C here, and one key for C here, and the next C is going to be over here. Right? SAL KHAN: Yeah. Twice as far. Yeah. VI HART: And the next C on the piano would have to be like-- SAL KHAN: So if piano manufacturers-- they innately made it based on a logarithmic scale, whether they knew it or not. VI HART: Yeah. Because we think about it on a logarithmic scale. SAL KHAN: They could have made it on a linear scale, and then the keys would just get fatter and fatter as we went to the right. VI HART: Oh, yeah. Fatter keys instead of making them farther apart. SAL KHAN: Someone should make that. VI HART: A fat key piano. SAL KHAN: A linear scale piano. Yes. VI HART: That would be awesome. But that's not how we think of pitch. SAL KHAN: No. It might be hard to play that. VI HART: It would be [? awesome. ?] SAL KHAN: And it's just not pitch. It also even be how we perceive a magnitude of the frequencies. Because we have the decibel scale, which is a logarithmic scale. VI HART: Yeah. There's a lot of natural, intuitive, logarithmic scales. So when we're looking at how loud something is, that's also the difference between how I'm talking now and how I'm talking if I'm a little louder. And we feel like distances between loudness also-- SAL KHAN: Right. VI HART: It's harder to explain. SAL KHAN: We perceive it a lot-- it is harder to explain. But we'll leave it there. VI HART: I don't have any pictures of how loud something is. SAL KHAN: We don't. No. And we don't want to bother people by just getting louder and louder. VI HART: By screaming. SAL KHAN: Yes. VI HART: I can scream at you some more. I think that would be great. SAL KHAN: Right. No. But that's fascinating. Especially this little game here. I'm going to start doing this at the next party I go to. VI HART: Mhm. It's good. And it makes sense. When we're looking at things, the difference between how much $1 million and $10 million is. SAL KHAN: Yes. VI HART: The world follows these kind of rules. SAL KHAN: Right, right, right. Very cool. VI HART: Yeah. SAL KHAN: Awesome.